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GMAT/Data Sufficiency - Median Matters
Median Matters
If x and y are unknown positive integers, is the mean of the set {6, 7, 1, 5, x, y} greater than the median of the set?
(1) x + y = 7
(2) x – y = 3
(A) Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) alone is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.
| ОТВЕТЫ И КОММЕНТАРИИ |
| Решение |
To find the mean of the set {6, 7, 1, 5, x, y}, use the average formula:
 where A = the average, S = the sum of the terms, and n = the number of terms in the set.
Using the information given in statement (1) that x + y = 7, we can find the mean: 
Regardless of the values of x and y, the mean of the set is  because the sum of x and y does not change.
To find the median, list the possible values for x and y such that x + y = 7. For each case, we can calculate the median.
x | y | DATA SET | MEDIAN |
1 | 6 | 1, 1, 5, 6, 6, 7 | 5.5 |
2 | 5 | 1, 2, 5, 5, 6, 7 | 5 |
3 | 4 | 1, 3, 4, 5, 6, 7 | 4.5 |
4 | 3 | 1, 3, 4, 5, 6, 7 | 4.5 |
5 | 2 | 1, 2, 5, 5, 6, 7 | 5 |
6 | 1 | 1, 1, 5, 6, 6, 7 | 5.5 |
Regardless of the values of x and y, the median (4.5, 5, or 5.5) is always greater than the mean ( ).
Therefore, statement (1) alone is sufficient to answer the question.
Now consider statement (2). Because the sum of x and y is not fixed, the mean of the set will vary. Additionally, since there are many possible values for x and y, there are numerous possible medians. The following table illustrates that we can construct a data set for which x – y = 3 and the mean is greater than the median. The table ALSO shows that we can construct a data set for which x – y = 3 and the median is greater than the mean.
x | y | DATA SET | MEDIAN | MEAN |
22 | 19 | 1, 5, 6, 7, 19, 22 | 6.5 | 10 |
4 | 1 | 1, 1, 4, 5, 6, 7 | 4.5 | 4 |
Thus, statement (2) alone is not sufficient to determine whether the mean is greater than the median.
The correct answer is (A): Statement (1) alone is sufficient, but statement (2) alone is not sufficient.
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